Haku didn't have time to regenerate.
What don't you get about page 12? Kakashi had a flashback and says he wants to preserve the memory of Zabuza's and Haku's deaths, and Kabutoro is annoyed that they got defeated.
The ice mirror and sword over Haku were meant to provide symbolism. Blood falls from the sword onto Haku's cheek, to make it look as if Haku was crying, the same way it did the first time he died. The blood on the ice mirror was aligned with Zabuza's reflection such that it looked like Zabuza was crying, the same way it did when he was moved by Naruto's dialogue.
Edit: I actually felt this chapter made up for the way Kishimoto recklessly brought Zabuza and Haku back. It was as if Kakashi was taking the side of the readers when he showed his disapproval of ruining the moment captured in the end of the first arc.